Annihilation of exceptional points from different Dirac valleys in a 2D photonic system

Topological physics relies on Hamiltonian’s eigenstate singularities carrying topological charges, such as Dirac points, and – in non-Hermitian systems – exceptional points (EPs), lines or surfaces. So far, the reported non-Hermitian topological transitions were related to the creation of a pair of EPs connected by a Fermi arc out of a single Dirac point by increasing non-Hermiticity. Such EPs can annihilate by reducing non-Hermiticity. Here, we demonstrate experimentally that an increase of non-Hermiticity can lead to the annihilation of EPs issued from different Dirac points (valleys). The studied platform is a liquid crystal microcavity with voltage-controlled birefringence and TE-TM photonic spin-orbit-coupling. Non-Hermiticity is provided by polarization-dependent losses. By increasing the non-Hermiticity degree, we control the position of the EPs. After the intervalley annihilation, the system becomes free of any band singularity. Our results open the field of non-Hermitian valley-physics and illustrate connections between Hermitian topology and non-Hermitian phase transitions.

In these Supplementary Notes, we provide more details on the behavior of the Hermitian part of the Hamiltonian of our system. We demonstrate how the Hamiltonian parameters are extracted from the experiment, including the non-Hermiticity (linear dichroism). We present more details demonstrating the existence of exceptional points and their annihilation. Finally, we show that the winding of the real effective field constrains the possibility of a non-Hermitian topological transition with annihilation of the exceptional points: The annihilation is only possible if the total winding of the real effective field is zero.

I. TOPOLOGY OF THE HERMITIAN HAMILTONIAN: THEORY
In this section, we illustrate the topological transitions concerning the Hermitian part of the two-band Hamil-tonian (1) of the main text describing the intersection of cavity modes with different numbers. This Hamiltonian is shown below for convenience, together with the definitions: where m ′ 0 is the mass at the central frequency (between the two modes in question). β is the magnitude of the TE-TM spin orbit coupling. β ′ =h 2 (mV−mH) 2mHmV This Hermitian Hamiltonian can be written as a linear combination of identity and Pauli matrices which defines a real effective magnetic field Ω r acting on the polarization pseudospin. The two non-zero components of the field are Ω and Ω y r = −2βk x k y . The pseudospin of a given eigenstate is either aligned or anti-aligned with the effective field (so far as the Hamiltonian remains Hermitian). Experimentally, this pseudospin corresponds to the Stokes vector of light, cal-culated using the following expressions: where the Is are the intensities in each of the polarizations, namely, V=Vertical, H=Horizontal, D=Diagonal, A=Anti-diagonal, R=Right-handed circularly polarized, L=Left-handed circularly polarized. These intensities can be measured experimentally or calculated theoretically as the absolute value squared of the eigenstate wavefunction in the corresponding basis. Each pseudospin component is an average value of the corresponding spin operator represented by a Pauli matrix. The case β ′ = 0 has already been considered in 2,3 . The two parabolas are split at k = 0, with different possible signs of this splitting (band detuning).
For positive detuning, the TE-TM splitting compensates this splitting along k x leading to a linear crossing. On the contrary, the TE-TM splitting is changing sign along k y and adds to the k = 0 splitting. As a result, along k y the two parabola get away the one with respect to the other and do not cross. The resulting spectrum shows two tilted Dirac cones at ±k 0x . The winding number of the effective field around these two points is the same (+1), as shown in Fig. S1b showing the distribution of the pseudospin (and effective field) texture in the reciprocal space. This winding is inherited from the winding number 2 of the TE and TM modes. The Berry topological charge is +1/2. This topological phase remains while β ′ increases up to β ′ = β where two new crossing points between the parabola appear at k 0y which sets at infinity when the transition occurs. These crossing points correspond to tilted Dirac cones. The winding number of the in-plane pseudospin around these two points is (-1) and the Berry topological charge (-1/2), as shown on Figure S1b. The sum of the topological charges of the 4 Dirac points is zero and the bands are overall topologically trivial which allows the non-Hermitian topological transition we explore in the main text.
For negative detuning (Fig. S1c), the winding of the two Dirac points does not change, but their positions in reciprocal space are rotated by π/2 and the particular field distribution changes from monopolar to Rashba. The Dirac points appear along the Y axis because of the negative value of the detuning in the experiment, but their winding is that of the TE-TM field, as expected. These epxeriments were performed on the sample 2, in a region with an LC layer thickness of around 3.9 µm. Panels (c) of Fig. S2 and Fig. S3, demonstrating the experimentally extracted pseudospin texture exhibiting different total winding, are shown in Fig. 3 of the main text.

III. HAMILTONIAN PARAMETERS
The parameters of the Hermitian Hamiltonian (1) are extracted from the fit of the dispersion in the two perpendicular directions. The fits are shown in Fig. S4. In a given direction, the average of the two parabolicity coefficients (inverse masses) of the two polarizations, provides the value of 1/m x and 1/m y . Then, the difference between the two parabolicites in the two polarizations is β + β ′ for Y direction and β − β ′ for X direction. Fitting both allows to extract β and β ′ separately.  c Experimental and d simulated by Berreman method pseudospin texture in S1-S2 plane of lower energy band.

IV. NON-HERMITIAN PARAMETER
In Fig. S5, we show an example of a spectrum of transmission in two polarizations, clearly demonstrating different linewidths. The experimental points for each of the two polarizations are fitted with the Voigt function, as explained in Methods. We stress that while the non-Hermiticity of the Hamiltonian is a single constant, the linewidth at each particular wave vector is determined by the imaginary part of the corresponding eigenstate. Only at the positions of the Dirac points of the Hermitian Hamiltonian does the linewidth of the eigenstates correspond directly to the non-Hermiticity of the Hamiltonian δΓ. It is also the maximal possible imaginary part of the energy that could be observed for any wave vector. This is what we use to extract the non-Hermitian parameter δΓ.

V. EXCEPTIONAL POINTS AND THEIR ANNIHILATION
In this section, we present more information on exceptional points.
The 2D images shown in Fig. 4 of the main text do not  allow to indicate the uncertainties, which are important to prove that the exceptional points are indeed present in our system and that the transition associated with the annihilation of the EPs really takes place.
In Fig. S6a,b we plot the real and imaginary parts (respectively) of the complex energies extracted along the We also plot in Fig. S7 the real part of the energies in the crossing (EPs annihilated) and anti-crossing (EPs present) case. This sums up as a cross-section of Fig. 4a,d of the main text near the top right corner of the Fermi arc denoted by k ′ . E 0 is the difference between the real parts of the upper and lower branches, and is in general different for the two cases. In one case, this cross-section crosses an imaginary Fermi arc, and in the other case, a real Fermi arc that forms a full circle.
Most of the results shown in the main text and all results presented above in the Supplemental Notes were obtained with Sample 1. To demonstrate that the observed behavior is universal, and does not depend on a particular sample, we have performed extra measurements with Sample 2, characterized by different parameters.  The relevant parameters of the sample 2 are given below: β = 0.23 ± 0.01 meVµm 2 , β ′ = 0.35 ± 0.01 meVµm 2 , 2δΓ = 0.5 ± 0.07 meV.

VI. HERMITIAN WINDING AND ANNIHILATION OF EXCEPTIONAL POINTS
We consider a two-band system described by a Hamiltonian depending on a two-dimensional wave vector. The Hermitian part of the Hamiltonian is characterized by a presence of diabolical (Dirac) points. The two-band nature allows writing the Hamiltonian as a superposition of Pauli matrices. For many typical situations, such Hamiltonians can be written using only two Pauli matrices that we can choose to be σ x and σ y (massless Dirac Hamiltonian, Rashba and Dresselhaus spin-orbit couplings, TE-TM spin-orbit coupling for photons), whereas the third Pauli matrix σ z , responsible for a symmetry breaking, opens the gap at the diabolical points (the mass term in the Dirac Hamiltonian or the Zeeman splitting for electrons and photonic modes). In this case, it is possible to characterize each of the diabolical points and the whole reciprocal space by winding numbers, which, in turn, determine the topology (the Chern number) of the bands once the gap is opened.
We then consider the evolution of the Hamiltonian with an addition of non-Hermiticity, described by a single constant parameter (independent of the wave vector). Without loss of generality, we consider the non-Hermitian contribution to be described by the Pauli matrix σ x . In presence of the non-Hermiticity, each of the diabolical points gives rise to two exceptional points.
Thanks to the use of the Pauli matrices, it is possible to map the system's Hamiltonian to a spin-1/2 in a magnetic field, using the so-called pseudospin formalism, where the terms of the Hamiltonian are considered as an effective field. In the presence of a non-Hermiticity, this effective field has both real and imaginary components, Ω = Ω ′ + iΩ ′′ . An exceptional point requires the real and imaginary parts of the effective field to be equal to each other in magnitude and perpendicular in direction: Ω ′ = Ω ′′ and Ω ′ ⊥ Ω ′′ .
Since in our case the direction of the imaginary field is fixed Ω ′′ = Ω ′′ e x , and the real field is in the XY plane, the exceptional points can only be found where the xcomponent of the real field is zero Ω ′ = Ω ′ e y ⇔ Ω ′ x = 0. This condition constrains their evolution in the reciprocal space when the imaginary field is increased Ω ′′ → ∞ (or the real field is decreased Ω ′ → 0). The location of the ensemble of the points Ω ′ x = 0 is determined by the winding of the real field at the diabolical points. With the increase of Ω ′′ , the exceptional points are moving towards infinity. If no points, where the condition for the real part Ω ′ x = 0 is satisfied, can be found at infinity, then the exceptional points must annihilate for some finite critical value Ω ′′ c . Two cases are possible: 1. The overall winding of the real field is a non-zero integer. In this case, the orientation angle θ r of the real field at infinity changes at least once between θ r = 0 and θ r = 2π, necessarily going through π/2 and 3π/2, where Ω ′ x = 0. Therefore, the excep- tional points can be found for any value of Ω ′′ up to infinity, and thus they cannot annihilate (or if they annihilate accidentally, they must then reappear when Ω ′′ is increased).
2. The overall winding of the real field is zero. In this case, the orientation angle of the real field θ r at infinity is constrained to an interval which may not include π/2 or 3π/2. In this case, the exceptional points cannot be present at infinity, and therefore they must annihilate for a finite value of Ω ′′ c .
The two situations are illustrated in Fig. S9. Panel a shows the case with total winding 2 with a dipolar texture typical for the TE-TM field. The trajectories of the exceptional points are shown with red and blue lines, which are defined by the condition Ω ′ x = 0 (vertical orientation of the real field). At a large scale, the relative position of the two diabolical points does not play any role and the texture is given by θ r = 2φ (where φ is the polar angle of the wave vector). The exceptional points at infinity are therefore necessarily located at φ = ±π/4, ±3π/4 (while initially they move from the diabolical points in the vertical direction).
The second situation is illustrated by Fig. S9b, where the overall winding of the real field is zero. The angle of the real field at infinity is θ r = 0 and there can be no exceptional points. The condition Ω ′ x = 0 (vertical real field) gives the trajectories of the exceptional points shown in the figure, with the annihilation points marked with crosses.
Finally, the winding of the exceptional points themselves is shown with a color of the arrows (red and blue).
The winding of an exceptional point is the winding of the phase of the complex eigenenergies. Indeed, the energy of a second-order exceptional point is given by where w = ±1 is precisely the winding number and q is the absolute value of the parameter controlling the deviation from the exceptional point, for example, the wave vector (measured from the exceptional point). This eigenenergy can be obtained as a solution for different Hamiltonians, with different variation of real and imaginary effective fields. Below, we show that when the imaginary effective field Ω ′′ is constant, the winding of the exceptional point can be easily determined from the texture of the real effective field Ω ′ . The most general form of a Hamiltonian of such type iŝ H = αq (σ x cos φ + iwσ y sin φ) (S6) + a((σ x cos θ + σ y sin θ) + ia (σ x cos(θ ± π/2) + σ y sin(θ ± π/2)) where the a gives the values of the real and imaginary effective field oriented at an angle θ and θ + π/2 respectively, equal and perpendicular at the exceptional point, w is the winding of the parameter-dependent part of the real effective field and α is its strength. The eigenenergies of this Hamiltonian are given by E ≈ ± √ q √ 2αae ±iwφ/2 e −iθ/2 (S7) We see that the winding of an exceptional point is determined by the winding w of the original diabolical point, modified by the relative angle of the real and imaginary fields at each exceptional point ±π/2.